\begin{answer}
    Note that
    $$
    \begin{aligned}
        D_{KL}(\hat P\|P_\theta) &=  \sum_{x\in\cal X} \hat P(x)\log \frac{\hat P(x)}{P_\theta(x)}\\
        &= \sum_{x\in\cal X}\hat P(x)\log\hat P(x) - \sum_{x\in\cal X}\hat P(x)\log P_\theta(x)\\
        &= c - \sum_{i=1}^m \log P_\theta(x)
    \end{aligned}
    $$

    And minimizing this is just maximizing the log likelihood. 
\end{answer}
